Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex (computationally) it is to use these logics to actually test whether a given graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal µ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL * with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP) complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic.

Communicated by C.A. Weibel To Alcides Lins Neto on his 60th birthday MSC: Primary: 13P10 37F75 secondary: 32S65 34M45 a b s t r a c tWe present an algorithm that can be used to check whether a given derivation of the complex affine plane has an invariant algebraic curve and discuss the performance of its implementation in the computer algebra system Singular.

Graphs are among the most frequently used structures in Computer Science. In this work, we analyze how we can express some important graph properties such as connectivity, acyclicity and the Eulerian and Hamiltonian properties in a modal logic. First, we show that these graph properties are not definable in a basic modal language. Second, we discuss an extension of the basic modal language with fix-point operators, the modal μ-calculus. Unfortunately, even with all its expressive power, the μ-calculus fails to express these properties. This happens because μ-calculus formulas are invariant under bisimulations. Third, we show that it is possible to express some of the above properties in a basic hybrid logic. Fourth, we propose an extension of CTL * with nominals, that we call hybrid-CTL * , and then show that it can express the Hamiltonian property in a better way than the basic hybrid logic. Finally, we introduce a promising way of expressing properties related to edges and use it to express the Eulerian property.

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In 1997, Goldreich, Goldwasser and Halevi presented the GGH cryptosystem, which is based on hard lattice problems. Only two years later, Nguyen pointed out major flaws on the scheme. From that point on, the system was considered officially dead. However, in 2012, Yoshino and Kunihiro proposed some improvements on the GGH cryptosystem, claiming to have fixed the flaws pointed out by Nguyen. In this paper, we make a thorough analysis of this tweaked GGH scheme, showing that, in practice, it behaves mostly in the same way as the original scheme. We also propose some modifications that can effectively make the new GGH different from the original one.